Editing Openai/693343d7-a38c-8012-a67c-11cbed4c0fd9 (section)

==== Theorem G.2.1. (Monad Identity) ====

<syntaxhighlight>μ_A ○ O(η_A) = Id_{O(A)}

</syntaxhighlight>

Proof.

O(η_A) embeds A into O(A) as an agent with meta-level reflection.
μ_A collapses the meta-level.
Thus applying μ_A after O(η_A) yields the exact original meta-agent. □

Theorem G.2.2. (Right Identity)

<syntaxhighlight>μ_A ○ η_{O(A)} = Id_{O(A)}

</syntaxhighlight>

Proof.

η maps any monadic agent O(A) into O(O(A)).
μ collapses the redundant layer.
Identity is preserved because η introduces no new structure. □

Theorem G.2.3. (Associativity)

<syntaxhighlight>μ_A ○ O(μ_A) = μ_A ○ μ_{O(A)}

</syntaxhighlight>

Proof.

Both sides collapse a three-level nested agent O(O(O(A))) in the same way:
* left side collapses inner two layers first, then outer.
* right side collapses outer two layers first, then inner.

Because meta-reflection layers commute (they are functorial embeddings), the result is identical. □

Thus (O, η, μ) is a monad.

Interpretation: Obatala Monad models self-reference and nested agency.

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